How are the Bellman optimality equations and minimax related?
They have similar philosophies, in the sense that minimax and algorithms based on the Bellman optimality equations are used to solve optimization problems, but they are also different because they solve different problems. 

Minimax (at least, the minimax version that I am aware of) is typically used to solve two-player games (e.g. chess, tic-tac-toe, etc.), while Bellman optimality equations (I assume you are referring to the Bellman equations that algorithms such as policy iteration are based on) do not assume the existence of two players (unless you consider the environment a player).